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6 R. MeyerA functor between symmetric monoidal categories is called symmetric monoidal ifit is compatible with the tensor products in a suitable sense [52]. A trivial example isthe functor W C alg ! G-C alg that equips a C -algebra with the trivial G-action.It follows from the universal property that ˝max is compatible with full crossedproducts: if A 22 G-C alg, B 22 C alg, then there is a natural isomorphismG Ë A ˝max .B/ Š .G Ë A/ ˝max B: (3)Like full crossed products, the maximal tensor product may be hard to describe becauseit involves a maximum of all possible C -tensor norms. There is another C -tensor normthat is defined more concretely and that combines well with reduced crossed products.Recall that any C -algebra A can be represented faithfully on a Hilbert space. Thatis, there is an injective -homomorphism A ! B.H/ for some Hilbert space H; hereB.H/ denotes the C -algebra of bounded operators on H. IfA is separable, we can findsuch a representation on the separable Hilbert space H D `2.N/. The tensor product oftwo Hilbert spaces H 1 and H 2 carries a canonical inner product and can be completedto a Hilbert space, which we denote by H 1 x˝ H 2 . If H 1 and H 2 support faithfulrepresentations of C -algebras A 1 and A 2 , then we get an induced -representation ofA 1 ˝ A 2 on H 1 x˝ H 2 .Definition 12. The minimal tensor product A 1 ˝min A 2 is the completion of A 1 ˝ A 2with respect to the operator norm from B.H 1 x˝ H 2 /.It can be check that this is well-defined, that is, the C -norm on A 1 ˝ A 2 does notdepend on the chosen faithful representations of A 1 and A 2 . The same argument alsoyields the naturality of A 1 ˝min A 2 . Hence we get a bifunctor˝min W G-C alg G-C alg ! G-C algIit defines another symmetric monoidal category structure on G-C alg.We may also call A 1 ˝min A 2 the spatial tensor product. It is minimal in the sensethat it is dominated by any C -tensor norm on A 1 ˝A 2 that is compatible with the givennorms on A 1 and A 2 . In particular, we have a canonical surjective -homomorphismA 1 ˝max A 2 ! A 1 ˝min A 2 : (4)Definition 13. AC -algebra A 1 is nuclear if the map in (4) is an isomorphism for allC -algebras A 2 .The name comes from an analogy between nuclear C -algebras and nuclear locallyconvex topological vector spaces (see [20]). But this is merely an analogy: the onlyC -algebras that are nuclear as locally convex topological vector spaces are the finitedimensionalones.Many important C -algebras are nuclear. This includes the following examples:• commutative C -algebras;• matrix algebras and algebras of compact operators on Hilbert spaces;